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ROMANIAN JOURNAL OF INFORMATIONSCIENCE AND TECHNOLOGYVolume 18, Number 2, 2015, 144–165
Computing Topological Indicesof Honeycomb Derived Networks
Sakander HAYAT1, Mehar Ali MALIK2, Muhammad IMRAN2
1School of Mathematical Sciences, University of Scienceand Technology of China (USTC), Hefei, Anhui, China
E-mail: [email protected] of Mathematics, School of Natural Sciences (SNS),
National University of Sciences and Technology (NUST),Sector H-12, Islamabad, Pakistan
E-mail: {alies.camp,imrandhab}@gmail.com
Abstract. A topological index can be considered as a transformation of a
chemical structure into a real number. The degree based topological indices such
as Randic index, geometric-arithmetic (GA) index and atom-bond connectivity
(ABC) index are of vital importance among all topological indices. These topo-
logical descriptors significantly correlate certain physico-chemical properties of the
corresponding chemical compounds. Graph theory has been found to be very useful
in this area of research.
The topological indices of certain interconnection and mesh derived networks are
recently studied by Imran et al. [17]. In this paper, we define some new classes of
networks from honeycomb networks by using basic graph operations like stellation,
medial and dual of a graph. We derive analytical close formulas of general Randic
index Rα(G) (for different values of α) for hexagonal and honeycomb derived net-
works. We also compute first Zagreb, ABC, and GA indices for these important
classes of networks.
Key-words: General Randic index, First Zagreb index, Atom-bond connectiv-
ity (ABC) index, Geometric-arithmetic (GA) index, Honeycomb derived networks.
2010 Mathematics Subject Classification: 05C12, 05C90
Computing Topological Indices of Honeycomb Derived Networks 145
1. Introduction and preliminary results
A topological index is a function “Top” defined from∑
to the set of real numbers,where
∑is the set of all finite simple graphs such that for G,H ∈
∑, we have
Top(G) = Top(H) if G and H are isomorphic. A topological index is a numericquantity associated with the chemical constitution of a chemical compound aimingthe correlation of chemical structure with many of its physico-chemical properties,chemical reactivity or biological activities. Topological indices are designed on theground of transformation of a molecular graph into a number which characterize thetopology of that graph.
Motivated by the chemical significance of topological indices, a lot of research hasbeen done in this area and different graph families have been studied. For example, afixed interconnection parallel architecture is characterized by a graph, with verticescorresponding to processing nodes and edges representing communication links. Inter-connection networks are notoriously hard to compare in abstract terms. Researchersin parallel processing are thus motivated to propose new and improved interconnec-tion networks, arguing the benefits and offering performance evaluations in differentcontexts [6, 33]. A few networks such as hexagonal, honeycomb, and grid networks,for instance, bear resemblance to atomic or molecular lattice structures. These net-works have very interesting topological properties which have been studied in differentaspects in [1, 2, 7, 12, 17, 22, 26].
The hexagonal and honeycomb networks have also been recognized as crucial forevolutionary biology, in particular for the evolution of cooperation, where the overlap-ping triangles are vital for the propagation of cooperation in social dilemmas. Relevantresearch that applies this theory and which could benefit further from the insights ofthe new research is found in [24, 25, 29, 32].
A graph G(V,E) with vertex set V and edge set E is said to be connected, ifthere exist a connection between any pair of vertices in G. A network is a connectedgraph having no multiple edges and loops. A planar graph, usually represented asG(V,E, F ), where F is the set of all regions (or faces), is a graph which can be drawnin the plane without crossing any edge, and such a drawing of G is called its planedrawing. A chemical graph is a graph whose vertices denote atoms and edges denotechemical bonds between atoms of the underlying chemical structure.
A (u, v)-path of length l in G is a sequence of l+ 1 vertices and l edges, from u tov, where u, v ∈ V . The degree du of a vertex u ∈ V is the number of vertices whichare connected to u by an edge. In a chemical graph the degree of any vertex is atmost4. The distance between two vertices u and v is denoted as d(u, v) and is the shortestpath between u and v in G. The length of shortest path between u and v is also calledu − v geodesic. The longest path between any two vertices u, v ∈ V , is called u − vdetour.
Throughout this paper, G is considered to be a network with vertex set V (G),edge set E(G) and du is the degree of vertex u ∈ V (G). The notations used in thispaper are mainly taken from the books [8, 10].
146 S. Hayat et al.
The theory of topological indices has its origin in the work done by Harold Wienerin 1947 while he was working on boiling point of paraffin. He named this index aspath number. Later on, the term “path number” was renamed as Wiener index [31].
Definition 1.1. Let G be a graph. Then the Wiener index of G is defined as
W (G) =1
2
∑(u,v)
d(u, v),
where (u, v) is any ordered pair of vertices in G and d(u, v) is the length of an u− vgeodesic. One of the oldest degree based topological index is Randic index [27] whichwas introduced by Milan Randic in 1975.
Definition 1.2. The Randic index of graph G is defined as
R− 12(G) =
∑uv∈E(G)
1√dudv
.
The general Randic index was proposed by Bollobas and Erdos [4] and Amic et al.[3] independently, in 1998. Since then it has been extensively studied by both math-ematicians and theoretical chemists [16]. Many important mathematical propertieshave been established [5] and a survey of results can be found in [20].The general Randic index Rα(G) is the sum of (dudv)
α over all edges e = uv ∈ E(G)defined as
Rα(G) =∑
uv∈E(G)
(dudv)α. (1)
It can be seen that R− 12(G) is the particular case of Rα(G), when α = − 1
2 .An important topological index introduced about forty years ago by Gutman andTrinajstic [11] is the Zagreb index or more precisely first Zagreb index, defined asfollows.
Definition 1.3. Consider a graph G, then first Zagreb index is defined as
M1(G) =∑
uv∈E(G)
(du + dv). (2)
The second Zagreb index is defined in the following way.
Definition 1.4. Consider a graph G, then second Zagreb index is defined as
M2(G) =∑
uv∈E(G)
(du × dv). (3)
One of the well-known degree based topological indices is the atom-bond connec-tivity (ABC) index introduced by Estrada et al. in [9].
Computing Topological Indices of Honeycomb Derived Networks 147
Definition 1.5. For a graph G, the ABC index is defined as
ABC(G) =∑
uv∈E(G)
√du + dv − 2
dudv. (4)
Another well-known connectivity topological descriptor is geometric-arithmetic(GA) index which was introduced by Vukicevic et al. in [30].
Definition 1.6. Consider a graph G, then its GA index is defined as
GA(G) =∑
uv∈E(G)
2√dudv
(du + dv). (5)
2. Main results
In this paper, we construct some new structures derived from hexagonal and hon-eycomb networks by using some basic graph operations. We give closed analytic for-mulas of the general Randic Rα(G), first Zagreb M1(G), atom-bond connectivityABC(G) and geometric-arithmeticGA(G) indices for these hexagonal and honeycombderived networks. For further study of topological indices of various graph familiessee [13–15, 19, 21].
2.1. Honeycomb derived networks
There are numerous open problems suggested for various interconnection networks.To quote Stojmenovic [28]:
Designing new architectures remains an area of intensive investigation giventhat there is no clear winner among existing ones.
In hexagonal network HX(n), the parameter n is the number of vertices on eachside of the network, whereas for honeycomb network HC(n), n is the number ofhexagons between boundary and central hexagon. Topological indices of hexagonal,honeycomb and other related networks have been studied in [26].
If we add a vertex in each face of a planar graph G and then join it to all thevertices of the respective face, we get the stellation of G, denoted as St(G) [23]. Thedual of a planar graph G, denoted by Du(G), is a graph whose vertex set is the set offaces of G, where two vertices f ′ and g′ are joined in Du(G) by an edge e′ if the facesf and g share the edge e in graph G. Clearly the number of vertices of Du(G) is equalto the number of faces of G and the number of edges of Du(G) is equal to the numberof edges of G. Since every planar graph has exactly one unbounded face, by deletingthe vertex corresponding to the unbounded face in Du(G), we get the bounded dualof the graph G, denoted as Bdu(G). These two operations are explained in Fig. 2.
148 S. Hayat et al.
(a). Hexagonal network (b). Honeycomb network
Fig. 1. (a) A hexagonal network of dimension 4; (b) A 3-dimensional honeycomb network.
The graph G
Dual of G (dotted)
Bounded dual of G (dotted) Medial of G (dotted)
Stellation of G (dotted)
Fig. 2. Stellation, dual, bounded dual and medial of a graph G.
Computing Topological Indices of Honeycomb Derived Networks 149
In this section, we derive some new classes of networks from honeycomb networkby using some basic graph operations like stellation, bounded dual and medial of agraph. These operations are already defined and explained in previous section. It caneasily be observed that the bounded dual of n-dimensional honeycomb networks isthe n-dimensional hexagonal network. Table 1 shows derivation of honeycomb de-rived networks in which G is the honeycomb networks, St(G) is the stellation of Gand Bdu(G) is the bounded dual of G. We denote n-dimensional honeycomb derivednetwork of first type as HcDN1(n) and of second type as HcDN2(n).
Table 1. Derivation of honeycomb derived networks
G∪ St(G)∪ Bdu(G)∪ Md(G) Planar/Non-planar
HcDN1 X × × Planar
HcDN2 X X × Non-planar
HcDN3 X × X Non-planar
HcDN4 X X X Non-planar
Now we study the topological indices of honeycomb derived networks of first typei.e. HcDN1(n). A 3-dimensional HcDN1 network is depicted in Fig. 3, in whichblack colored graph is 3-dimensional honeycomb network and blue colored graph isits stellation.
Fig. 3. An HcDN1(n) network with n = 3.
Now we compute degree based topological indices of honeycomb derived (HcDN1)networks in the following theorems. First, we compute general Randic index Rα(G),with α = 1,−1, 12 ,−
12 .
Theorem 2.1.1. Consider the honeycomb derived network HcDN1(n), n ≥ 3,then its general Randic index is equal to
150 S. Hayat et al.
Rα(HcDN1(n)) =
972n2 − 1224n+ 414, α = 1;
162n2 + (12√
15 + 18√
2 + 18√
30−
342)n− 12√
15− 18√
30 + 234, α =1
2;
3
4n2 +
3
20n+
1
10, α = −1;
9
2n2 +
(4√
15
5+
3√
30
5+√
2− 19
2
)n−
4√
15
5− 3√
30
5+ 7, α = −1
2.
Proof. Let G be the HcDN1(n) network of dimension n. The number of verticesand edges in HcDN1(n) are 9n2 − 3n + 1 and 27n2 − 21n + 6, respectively. Thereare five types of edges in HcDN1(n) based on degrees of end vertices of each edge.Table 2 shows such an edge partition of HcDN1(n).
Table 2. Edge partition of honeycombderived network HcDN1(n), n ≥ 3 based on
degrees of end vertices of each edge
(du, dv) where uv ∈ E(G) Number of edges
(3, 3) 6
(3, 5) 12(n− 1)
(3, 6) 6n
(5, 6) 18(n− 1)
(6, 6) 27n2 − 57n+ 30
We consider the following cases for the possible values of α.Case 1. α = 1We apply the formula of Rα(G) given by equation (1) for α = 1. By using edgepartition given in Table 2, we get
R1(G) = 972n2 − 1224n+ 414.
Case 2. α =1
2
We apply the formula of Rα(G) given by equation (1) for α =1
2. By using edge
partition given in table 2, we get
R 12(G) = 162n2 + (12
√15 + 18
√2 + 18
√30− 342)n− 12
√15− 18
√30 + 234.
Case 3. α = −1We apply the formula of Rα(G) given by equation (1) for α = −1. By using edgepartition given in table 2, we get
R−1(G) =3
4n2 +
3
20n+
1
10.
Computing Topological Indices of Honeycomb Derived Networks 151
Case 4. α = −1
2
We apply the formula of Rα(G) given by equation (1) for α = −1
2. By using edge
partition given in Table 2, we get
R− 12(G) =
9
2n2 +
(4√
15
5+
3√
30
5+√
2− 19
2
)n− 4
√15
5− 3√
30
5+ 7.
ut
In the following theorem, we compute first Zagreb index of an n-dimensional hon-eycomb derived network of first type.
Theorem 2.1.2 For an n-dimensional honeycomb network HcDN1(n), n ≥ 3,the first Zagreb index is equal to
M1(HcDN1(n)) = 324n2 − 336n+ 102.
Proof. Let G be the HcDN1(n) network with n ≥ 3. By using the formula givenby equation (2) and the edge partition given in table 2, we get
M1(G) = 324n2 − 336n+ 102.
ut
Now we compute ABC index of honeycomb derived network HcDN1(n).
Theorem 2.1.3 Consider the honeycomb derived network HcDN1(n), n ≥ 3,then its ABC index is equal to
ABC(HcDN1(n)) =9√
10
2n2 +
(12√
10
5+
9√
30
5− 19
√10
2+ 14
)n− 12
√10
5−
−9√
30
5+ 5√
10 + 4.
Proof. Let G be the honeycomb derived network of first type. By using the formulagiven by equation (4) and using edge partition given in Table 2, we get
ABC(G) =9√
10
2n2 +
(12√
10
5+
9√
30
5− 19
√10
2+ 14
)n− 12
√10
5−
−9√
30
5+ 5√
10 + 4.
ut
In the following theorem, we compute GA index of honeycomb derived networkHcDN1(n).
152 S. Hayat et al.
Theorem 2.1.4 Consider the honeycomb derived network HcDN1(n), n ≥ 3,then its GA index is equal to
GA(G) = 27n2 +
(36√
30
11+ 3√
15 + 4√
2− 57
)n− 36
√30
11− 3√
15 + 36.
Proof. Let G be the honeycomb derived network of first type. By using the formulagiven by equation (5) and using the edge partition given in Table 2, we get
GA(G) = 27n2 +
(36√
30
11+ 3√
15 + 4√
2− 57
)n− 36
√30
11− 3√
15 + 36.
ut
Now we compute the above discussed degree based topological indices for the sec-ond type of honeycomb derived networks, that is HcDN2(n). An HcDN2(n) networkis defined as the union of honeycomb network, its stellation and its bounded dual asshown in Table 1. A 3-dimensional HcDN2 network is depicted in Fig. 4 in whichblack colored graph is a honeycomb network, blue colored graph is its stellation andthe red color represents its bounded dual.
Fig. 4. An HcDN2(n) network with n = 3.
In following theorems, we calculate the exact expressions for the general Randicindex Rα(G) for different values of α.
Theorem 2.1.5 Consider the honeycomb derived HcDN2(n) networks with n ≥3, then its general Randic index is equal to
Computing Topological Indices of Honeycomb Derived Networks 153
Rα(HcDN2(n)) =
2916n2 − 5136n+ 2310, α = 1;
(108√
2 + 162)n2 + (48√
15 + 36√
30−174√
2− 462)n− 84√
15 + 72√
3− 66√
30+
36√
5− 132√
2 + 36√
6 + 36√
10 + 270, α =1
2;
9
16n2 +
27
80n+
97
1800, α = −1;(
3√
2
2+
9
4
)n2 +
(7√
15
5+
3√
30
5− 33
√2
10−
113
20
)n− 2
√15 +
5√
3
3−√
30 +4√
5
5+
11√
2
10+
2√
6
3+
2√
10
5+
47
10, α = −1
2.
Proof. Let G be the HcDN2(n) network of dimension n, where n ≥ 3. The vertexand edge cardinalities of HcDN2(n) are 9n2−3n+1 and 27n2−21n+6 respectively.There are sixteen types of edges in an HcDN2(n) network, based on degrees of endvertices of each edge. Table 3 shows such an edge partition of HcDN2(n).
Table 3. Edge partition of honeycombderived network HcDN2(n), n ≥ 3 based on
degrees of end vertices of each edge
(du, dv) where uv ∈ E(G) Number of edges
(3, 3) 6
(3, 5) 12(n− 1)
(3, 9) 12
(3, 10) 6(n− 2)
(5, 6) 6(n− 1)
(5, 9) 12
(5, 10) 12(n− 2)
(6, 6) 9n2 − 21n+ 12
(6, 9) 12
(6, 10) 18(n− 2)
(6, 12) 18n2 − 54n+ 42
(9, 10) 12
(9, 12) 6
(10, 10) 6(n− 3)
(10, 12) 12(n− 2)
(12, 12) 9n2 − 33n+ 30
We consider the following cases for the possible values of α.Case 1. α = 1We apply the formula of Rα(G) given by equation (1) for α = 1. By using edge
154 S. Hayat et al.
partition given in Table 3, we get
R1(G) = 2916n2 − 5136n+ 2310.
Case 2. α =1
2
Using the formula of Rα(G) given by equation (1) for α =1
2and edge partition given
in Table 3, after some simplification, we get
R 12(G) = (108
√2 + 162)n2 + (48
√15 + 36
√30− 174
√2− 462)n− 84
√15 + 72
√3−
−66√
30 + 36√
5− 132√
2 + 36√
6 + 36√
10 + 270.
Case 3. α = −1The formula of Rα(G) is given by equation (1) for α = −1. By using edge partitiongiven in Table 3 and after some simplification, we get
R−1(G) =9
16n2 +
27
80n+
97
1800.
Case 4. α = −1
2
Using the formula of Rα(G) given by equation (1) for α = −1
2and the edge partition
given in Table 3, after some simplification, we get
R− 12(G) =
(3√
2
2+
9
4
)n2 +
(7√
15
5+
3√
30
5− 33
√2
10− 113
20
)n− 2
√15 +
5√
3
3−
−√
30 +4√
5
5+
11√
2
10+
2√
6
3+
2√
10
5+
47
10.
ut
In the following theorem, first Zagreb index of an n-dimensional honeycomb de-rived network of second type is computed.
Theorem 2.1.6. For an n-dimensional HcDN2(n) network with n ≥ 3, the firstZagreb index is equal to
M1(HcDN2(n)) = 648n2 − 924n+ 360.
Proof. Let G be the HcDN2(n) network with n ≥ 3. By using the formula givenby equation (2) and some easy calculations, we get
M1(G) = 648n2 − 924n+ 360.
ut
Now we compute ABC index of honeycomb derived network HcDN2(n).
Computing Topological Indices of Honeycomb Derived Networks 155
Theorem 2.1.7. Consider the honeycomb derived network HcDN2(n), n ≥ 3,then its ABC index is equal to
ABC(HcDN2(n)) =
(3√
10
2+
3√
22
4+ 6√
2
)n2 +
(12√
10
5+
√330
5+
3√
30
5+
+6√
26
5+
9√
2
5+
3√
210
5− 7√
10
2− 11
√22
4− 12
√10
5+
+4√
30
3− 2√
330
5− 3√
30
5+
8√
15
5− 18
√2 + 2
√6
)n−
−12√
26
5+
2√
78
3+
2√
170
30− 6√
210
5+
5√
22
2− 9√
2
5+
√57
3+
+2√
10 + 14√
2− 4√
6.
Proof. Let G be the honeycomb derived network of first type. By using the formulagiven by equation (4) and doing simplification, we get the following result:
ABC(HcDN2(n)) =
(3√
10
2+
3√
22
4+ 6√
2
)n2 +
(12√
10
5+
√330
5+
3√
30
5+
+6√
26
5+
9√
2
5+
3√
210
5− 7√
10
2− 11
√22
4− 12
√10
5+
+4√
30
3− 2√
330
5− 3√
30
5+
8√
15
5− 18
√2 + 2
√6
)n−
−12√
26
5+
2√
78
3+
2√
170
30− 6√
210
5+
5√
22
2− 9√
2
5+
√57
3+
+2√
10 + 14√
2− 4√
6.
ut
The following theorem exhibits GA index of honeycomb derived network of secondversion i.e. HcDN2(n).
Theorem 2.1.8. Consider the honeycomb derived network HcDN2(n), n ≥ 3,then its GA index is equal to
GA(HcDN2(n)) =
(12√
2 + 18
)n2 +
(600√
30
143+
9√
15
2+ 3√
15− 28√
2− 48
)n−
−1044√
30
143+
36√
5
7+
24√
6
5+
72√
10
19+
24√
3
7− 3√
15 + 6√
3−
−16√
2− 9√
15 + 28√
2 + 30.
Proof. Let G be the honeycomb derived network of first type. By using the formulagiven by equation (5) and the edge partition given in Table 3, after simplificationwe get:
156 S. Hayat et al.
GA(HcDN2(n)) =
(12√
2 + 18
)n2 +
(600√
30
143+
9√
15
2+ 3√
15− 28√
2− 48
)n−
−1044√
30
143+
36√
5
7+
24√
6
5+
72√
10
19+
24√
3
7− 3√
15 + 6√
3−
−16√
2− 9√
15 + 28√
2 + 30.
utNow we study above discussed topological indices for third type of honeycomb
derived networks, that is HcDN3(n). An HcDN3(n) network is defined as a union ofhoneycomb network, its stellation and its medial. The vertex and edge cardinalities ofaHcDN3(n) network are 18n2−6n+1 and 54n2−42n+12 respectively. AnHcDN3(3)is depicted in Fig. 5, in which green colored graph is the medial of honeycomb network.
Fig. 5. An HcDN3(n) network with n = 3.
Now we compute general Randic index for different values of α.
Theorem 2.1.9.Consider the honeycomb derived HcDN3(n) networks withn ≥ 3, then its general Randic index is equal to:
Rα(HcDN3(n)) =
1944n2 − 2472n+ 876, α = 1;
324n2 + (24√
3 + 18√
2 + 18√
30+
24√
5 + 24√
6− 624)n− 24√
5−−24√
6− 18√
30 + 324, α =1
2;
3
2n2 +
49
120n− 1
5, α = −1;
9n2 +(6√
5
5+
3√
30
5+ 2√
3 +√
2 +√
6− 33
2
)n−
−6√
5
5− 3√
30
5−√
6− 9, α = −1
2.
Computing Topological Indices of Honeycomb Derived Networks 157
Proof. Let G be the HcDN3(n) network of dimension n. There are seven typesof edges in HcDN3(n) based on degrees of end vertices of each edge. Table 4 showssuch an edge partition of HcDN3(n).
Table 4. Edge partition of honeycombderived network HcDN3(n), n ≥ 3 based on
degrees of end vertices of each edge
(du, dv) where uv ∈ E(G) Number of edges
(3, 4) 12n
(3, 6) 6n
(4, 4) 6n
(4, 5) 12(n− 1)
(4, 6) 12(n− 1)
(5, 6) 18(n− 1)
(6, 6) 54n2 − 108n+ 54
We consider the following cases for the possible values of α.Case 1. α = 1We apply the formula of Rα(G) given by equation (1) for α = 1. By using edgepartition given in Table 4 and after simplifying, we get
R1(G) = 1944n2 − 2472n+ 876.
Case 2. α =1
2We apply the formula of Rα(G) given by equation (1) for α =
1
2. By using edge
partition given in Table 4, some simplification gives
R 12(G) = 324n2 + (24
√3 + 18
√2 + 18
√30 + 24
√5 + 24
√6− 624)n−
24√
5− 24√
6− 18√
30 + 324.
Case 3. α = −1We apply the formula of Rα(G) given by equation (1) for α = −1. By using edgepartition given in Table 4, then after simplification we get
R−1(G) =3
2n2 +
49
120n− 1
5.
Case 4. α = −1
2We apply the formula of Rα(G) given by equation (1) for α = −1
2. By using edge
partition given in Table 4, then after simplification we get
R− 12(G) = 9n2 +
(6√
5
5+
3√
30
5+ 2√
3 +√
2 +√
6− 33
2
)n− 6
√5
5− 3√
30
5−√
6− 9.
ut
158 S. Hayat et al.
In the following theorem, we compute first Zagreb index of an n-dimensional hon-eycomb derived network of third type.
Theorem 2.1.10.For an n-dimensional HcDN3(n) network with n ≥ 3, thenfirst Zagreb index is equal to:
M1(HcDN3(n)) = 648n2 − 684n+ 222.
Proof. Let G be the HcDN3(n) network with n ≥ 3. By using the formula givenby equation (2) and using edge partition given in Table 4 and after simplifications weget
M1(G) = 648n2 − 684n+ 222.
utNow we compute ABC index of honeycomb derived network HcDN3(n).
Theorem 2.1.11.Consider the honeycomb derived network HcDN3(n), n ≥ 3,then its ABC index is equal to:
ABC(HcDN3(n)) = 9√
10n2 +
(6√
35
5+
9√
30
5+
3√
6
2+ 2√
15 + 4√
3 +√
14−
−18
)n− 6
√35
5− 9√
30
5+ 9√
10− 4√
3.
Proof. Let G be the honeycomb derived network of third type. By using the for-mula given by equation (4) and by using edge partition given in Table 4 and somesimplifications, we get:
ABC(HcDN3(n)) = 9√
10n2 +
(6√
35
5+
9√
30
5+
3√
6
2+ 2√
15 + 4√
3 +√
14−
−18
)n− 6
√35
5− 9√
30
5+ 9√
10− 4√
3.
utIn the following theorem, we compute GA index of honeycomb derived network
HcDN3(n).
Theorem 2.1.12.Consider the honeycomb derived network HcDN3(n), n ≥ 3,then its GA index is equal to:
GA(G) = 54n2 +
(48√
3
7+
24√
6
5+
16√
5
3+
36√
30
11+ 4√
2− 102
)n− 16
√5
3−
−24√
6
5− 36
√30
11+ 54.
Proof. LetG be the honeycomb derived network of third type. By using the formulagiven by equation (1) and using edge partition given in Table 4 and doing some
Computing Topological Indices of Honeycomb Derived Networks 159
simplifications, we get:
GA(G) = 54n2 +
(48√
3
7+
24√
6
5+
16√
5
3+
36√
30
11+ 4√
2− 102
)n−
−16√
5
3− 24
√6
5− 36
√30
11+ 54.
utNow we study the topological indices of the fourth type of honeycomb derived
networks, that is HcDN4(n). An HcDN4(n) network is defined as the union ofhoneycomb network, its stellation, its medial and its bounded dual, as shown in Table1. A 3-dimensional HcDN4 network is depicted in Fig. 6, in which black colored graphis honeycomb network, blue colored graph is its stellation, green color is its medialand red color represents its bounded dual. This network has 18n2 − 6n + 1 verticesand 63n2 − 57n+ 18 edges.
Vertices of this type are of degree 4,because the starting and ending verticesof red edges are central verticesof hexagons.
Fig. 6. An HcDN4(n) network with n = 3.
In the next theorem, we compute exact expressions for the general Randic index
Rα(G), for α = 1,1
2,−1,−1
2.
Theorem 2.1.13.Consider the honeycomb derived network HcDN4(n) with n ≥3, then its general Randic index is equal to:
160 S. Hayat et al.
Rα(HcDN4(n)) =
3888n2 − 6384n+ 2772, α = 1;
(108√
2 + 324)n2 + (24√
3 + 36√
30 + 24√
5 + 24√
6+
+36√
15− 264√
2− 744)n+ 72√
3− 66√
30− 24√
5+
+36√
5− 120√
2− 72√
15 + 252√
2 + 36√
10 + 396, α =1
2;
21
16n2 +
143
240n− 443
1800, α = −1;(
3√
2
2+
27
4
)n2 +
(3√
15
5+
6√
5
5− 33
√2
10+
3√
30
5+
+2√
3 +√
6− 253
20
)n+
5√
3
3−√
30− 2√
5
5−
−11√
2
10+
2√
10
5− 6√
15
5+
67
10, α = −1
2.
Proof. Let G be the HcDN4(n) network of dimension n with n ≥ 3. There areeighteen types of edges in HcDN4(n), based on the degrees of end vertices of eachedge which make a partition of edge set of G. Table 5 shows such an edge partitionof HcDN4(n).
Table 5. Edge partition of honeycombderived network HcDN4(n), n ≥ 3 based on
degrees of end vertices of each edge.
(du, dv) where uv ∈ E(G) Number of edges
(3, 4) 12n
(3, 9) 12
(3, 10) 6(n− 2)
(4, 4) 6n
(4, 5) 12(n− 1)
(4, 6) 12(n− 1)
(5, 6) 6(n− 1)
(5, 9) 12
(5, 10) 12(n− 2)
(6, 6) 36n2 − 72n+ 36
(6, 9) 12
(6, 10) 18(n− 2)
(6, 12) 18n2 − 54n+ 42
(9, 10) 12
(9, 12) 6
(10, 10) 6(n− 3)
(10, 12) 12(n− 2)
(12, 12) 9n2 − 33n+ 30
We consider the following cases for the possible values of α.
Case 1. α = 1
Computing Topological Indices of Honeycomb Derived Networks 161
Using the formula of Rα(G) given by equation (1) for α = 1 and the edge partitiongiven in Table 5, we get
R1(G) = 3888n2 − 6384n+ 2772.
Case 2. α =1
2.
Using the formula of Rα(G) given by equation (1) for α =1
2and using edge partition
given in Table 5, we get
R 12(G) =
(3√
2
2+
27
4
)n2 +
(3√
15
5+
6√
5
5− 33
√2
10+
3√
30
5+ 2√
3 +√
6−
−253
20
)n+
5√
3
3−√
30− 2√
5
5− 11
√2
10+
2√
10
5− 6√
15
5+
67
10.
Case 3. α = −1.Applying the formula of Rα(G) given by equation (1) for α = −1 and by using edgepartition given in Table 5, we get
R−1(G) =21
16n2 +
143
240n− 443
1800.
Case 4. α = −1
2.
Applying the formula of Rα(G) given by equation (1) for α = −1
2and using edge
partition given in Table 5, we get
R− 12(G) =
(3√
2
2+
9
4
)n2 +
(7√
15
5+
3√
30
5− 33
√2
10− 113
20
)n− 2
√15 +
5√
3
3−
−√
30 +4√
5
5+
11√
2
10+
2√
6
3+
2√
10
5+
47
10.
ut
In the following theorem, we compute first Zagreb index of an n-dimensional hon-eycomb derived network of fourth type.
Theorem 2.1.14.For an n-dimensional HcDN4(n) network with n ≥ 3, the firstZagreb index is equal to:
M1(HcDN4(n)) = 972n2 − 1272n+ 480.
Proof. Let G be the HcDN4(n) network with n ≥ 3. By using the formula givenby equation (2) and the edge partition from Table 5, we get:
M1(G) = 972n2 − 1272n+ 480.
ut
162 S. Hayat et al.
Now we compute ABC index of honeycomb derived network HcDN4(n).
Theorem 2.1.15.Consider the honeycomb derived network HcDN4(n), n ≥ 3,then its ABC index is equal to:
ABC(HcDN4(n)) =
(3√
22
4+ 6√
2 + 6√
10
)n2 +
(7√
6
2+
6√
35
5+
3√
30
5+
+6√
26
5+
3√
210
5− 81
√2
5− 11
√22
4+
√330
5+ 2√
15 +
+4√
3− 12√
10
)n+
4√
30
3− 2√
330
5− 6√
35
5− 3√
30
5+
+8√
15
5− 12
√26
5+
2√
78
3+
2√
170
5− 6√
210
5+
5√
22
2+
+61√
2
5+
√57
3− 4√
3 + 6√
10− 4√
6.
Proof. Let G be the honeycomb derived network of fourth type. By using theformula given by equation (4) and using edge partition given in Table 5, we get:
ABC(HcDN4(n)) =
(3√
22
4+ 6√
2 + 6√
10
)n2 +
(7√
6
2+
6√
35
5+
3√
30
5+
+6√
26
5+
3√
210
5− 81
√2
5− 11
√22
4+
√330
5+ 2√
15 +
+4√
3− 12√
10
)n+
4√
30
3− 2√
330
5− 6√
35
5− 3√
30
5+
+8√
15
5− 12
√26
5+
2√
78
3+
2√
170
5− 6√
210
5+
5√
22
2+
+61√
2
5+
√57
3− 4√
3 + 6√
10− 4√
6.
ut
In the following theorem, we compute GA index of honeycomb derived networkHcDN4(n).
Theorem 2.1.16.Consider the honeycomb derived network HcDN4(n), n ≥ 3,then its GA index is equal to:
GA(HcDN4(n)) =
(12√
2 + 45
)n2 +
(48√
3
7+
600√
30
143+
16√
5
3+
24√
6
5+
+9√
15
2− 28
√2− 93
)n− 1044
√30
143− 4√
5
21+
72√
10
19+
+66√
3
7− 9√
15 + 12√
2 + 48.
Computing Topological Indices of Honeycomb Derived Networks 163
Proof. Let G be the honeycomb derived network of fourth type. By using theformula given by equation (5) and using edge partition given in Table 5, we get:
GA(G) = 12n
(2√
3× 4
3 + 4
)+12
(2√
3× 9
3 + 9
)+(6n−12)
(2√
3× 10
3 + 10
)+6n
(2√
4× 4
4 + 4
)+
(12n−12)
(2√
4× 5
4 + 5
)+(12n−12)
(2√
4× 6
4 + 6
)+(6n−6)
(2√
5× 6
5 + 6
)+12
(2√
5× 9
5 + 9
)+
(12n − 24)
(2√
5× 10
5 + 10
)+ (36n2 − 72n + 36)
(2√
6× 6
6 + 6
)+ 12
(2√
6× 9
6 + 9
)+ (18n −
−36)
(2√
6× 10
6 + 10
)+(18n2−54n+42)
(2√
6× 12
6 + 12
)+12
(2√
9× 10
9 + 10
)+6
(2√
9× 12
9 + 12
)+
(6n− 18)
(2√
10× 10
10 + 10
)+ (12n− 24)
(2√
10× 12
10 + 12
)+ (9n2− 33n+ 30)
(2√
12× 12
12 + 12
).
After simplification, it gives
GA(HcDN4(n)) =
(12√
2 + 45
)n2 +
(48√
3
7+
600√
30
143+
16√
5
3+
24√
6
5+
+9√
15
2− 28
√2− 93
)n− 1044
√30
143− 4√
5
21+
72√
10
19+
+66√
3
7− 9√
15 + 12√
2 + 48.
ut
3. Conclusion and general remarks
In this paper, certain degree based topological indices, namely general Randicindex (Rα), atom-bond connectivity index (ABC), geometric-arithmetic index (GA)and first Zagreb index (M1) for hexagonal and honeycomb derived networks werestudied for the first time. We defined some new classes of networks derived fromthe already existing hexagonal and honeycomb networks by using some basic graphoperations. We computed analytical close formulas of the above mentioned degreebased topological indices for these derived networks. These results provide a basisto understand deep topology of these important networks. The reader is encouragedto design some new architectures/networks and study their topological indices tounderstand their topologies.
Acknowledgments. The first author is supported by CAS-TWAS president fel-lowship at University of Science and Technology of China (USTC), China. The secondand third authors are supported by National University of Sciences and Technology,Islamabad, Pakistan and by the grant of Higher Education Commission of PakistanRef. No. 20-367/NRPU/R&D/HEC/12/831.
164 S. Hayat et al.
References
1. BACA M., HORVATHOVA J., MOKRISOVA M., SUHANYIOVA A., On topologicalindices of fullerenes, Appl. Math. Comput. 251(2015), 154–161.
2. BAIG A. Q., IMRAN M., ALI H., On topological indices of poly oxide, poly silicate,DOX and DSL networks, Canad. J. Chem., DOI:10.1139/cjc-2014-0490.
3. AMIC D., BESLO D., LUCIC B., NIKOLIC S., TRINAJSTIC N., The vertex-connectivity index revisited, J. Chem. Inf. Comput. Sci. 38(1998), 819–822.
4. BOLLOBAS B., ERDOS P., Graphs of extremal weights, Ars Combinatoria 50(1998),225–233.
5. Caporossi G., Gutman I., Hansen P., Pavlovıc L., Graphs with maximum connectivityindex, Comput. Bio. Chem. 27(2003), 85–90.
6. CHEN M. S., SHIN K. G., KANDLUR D. D., Addressing, routing and broadcasting inhexagonal mesh multiprocessors, IEEE Trans. Comput. 39(1990), 10–18.
7. CYNTHIA V. J. A., Metric dimension of certain mesh derived graphs, J. Comp. Math.Sci. 1(2014), 71–77.
8. DIUDEA M. V., GUTMAN I., LORENTZ J., Molecular Topology, Nova, Huntington,2001.
9. ESTRADA E., TORRES L., RODRIGUEZ L., GUTMAN I., An atom-bond connectiv-ity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem. 37A(1998),849–855.
10. GUTMAN I., POLANSKY O. E., Mathematical concepts in organic chemistry,Springer-Verlag, New York, 1986.
11. GUTMAN I., TRINAJSTIC N., Graph theory and molecular orbitals, Total π-electronenergy of alternant hydrocarbons, Chem. Phys. Lett., 17(1972), 535–538.
12. HAYAT S., IMRAN M., Computation of topological indices of certain networks, Appl.Math. Comput. 240(2014), 213–228.
13. HAYAT S., IMRAN M., Computation of certain topological indices of nanotubes, J.Comput. Theor. Nanosci. 12(1)(2015), 70–76.
14. HAYAT S., IMRAN M., Computation of certain topological indices of nanotubes coveredby C5 and C7, J. Comput. Theor. Nanosci. 12(4)(2015), 533–541.
15. HAYAT S., IMRAN M., On topological properties of carbon nanocones CNCk[n], StudiaUBB Chemia, LIX, 3(2014), 113–128.
16. HU Y., LI X., SHI Y., XU T., GUTMAN I., On molecular graphs with smallest andgreatest zeroth-order general Randic index, MATCH Commun. Math. Comput. Chem.54(2005), 425–434.
17. IMRAN M., HAYAT S., MALIK M. Y. H., On topological indices of certain intercon-nectinon networks, Appl. Math. Comput. 244(2014), 936–951.
18. IMRAN M., HAYAT S., SHAFIQ M. K., On topological indices of nanostar dendrimersand polymino chains, Optoelectron. Adv. Mater. Rapid Commun. 8(9−10)(2014), 948–954.
19. IRANMANESH A., ZERAATKAR M., Computing GA index for some nanotubes, Op-toelectron. Adv. Mater. Rapid Commun. 4(2010), 1852–1855.
20. LI X., GUTMAN I., Mathematical aspects of Randic-type molecular structure descrip-tors, mathematical chemistry monographs No. 1, Kragujevac (2006).
21. MALIK M. A., HAYAT S., IMRAN M., On the anti-Kekule number of nanotubes andnanocones, J. Comput. Theor. Nanosci. Accepted, in press.
22. MANUEL P. D., RAJAN B., RAJASINGH I., MONICA M. C., On minimum metricdimension of honeycomb networks, J. Discrete Algorithms, 6(2008), 20–27.
Computing Topological Indices of Honeycomb Derived Networks 165
23. PISANSKI T., RANDIC M., In geometry at work, M. A. A. Notes 53(2000), 174–194.24. PERC M., GOMEZ-GARDENES J., SZOLNOKI A., FLORIA L. M., MORENO Y.,
Populations: a review: Evolutionary dynamics of group interactions on structured, J.R. Soc. Interface, 10(2013), 20120997.
25. PERC M., SZOLNOKI A., Coevolutionary games-A mini review, BioSystems,99(2010), 109–125.
26. RAJAN B., WILLIAM A., GRIGORIOUS C., STEPHEN S., On Certain TopologicalIndices of Silicate, Honeycomb and Hexagonal Networks, J. Comp. Math. Sci., 5(2012),530–535.
27. RANDIC M., On Characterization of molecular branching, J. Amer. Chem. Soc.97(1975), 6609–6615.
28. STOJMENOVIC I., Direct interconnection networks, in: A.Y. Zomaya (Ed.), Paralleland distributed computing handbook, (1996), pp. 537–567.
29. SZOLNOKI A., PERC M., SZABO G., Topology-independent impact of noise on coop-eration in spatial public goods games, Phys. Rev. E, 80(2009) 056109.
30. VUKICEVIC D., FURTULA B., Topological index based on the ratios of geometricaland arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46(2009), 1369–1376.
31. WIENER H., Structural determination of paraffin boiling points, J. Amer. Chem. Soc.69(1947), 17–20.
32. WANG Z., SZOLNOKI A., PERC M., If players are sparse social dilemmas are too:Importance of percolation for evolution of cooperation, Sci. Reports, 2(2012), 369.
33. XU J., Topological structure and analysis of interconnection networks, Kluwer AcademicPublishers, 2001.
